×

A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II: Explicit method. (English) Zbl 0598.65054

From authors’ summary: In part I [ibid. 11, 277-281 (1984; Zbl 0565.65041)] the authors gave a Noumerov-type method with minimal phase- lag for the integration of second order initial-value problems: \(y''=f(t,y)\), \(y(t_ 0)=y_ 0\), \(y'(t_ 0)=y_ 0'\). However, the method given there is implicit. We show here the interesting result that if the Noumerov-type methods of the paper mentioned above are made explicit with the help of the classical second order method, then there exists a selection of the free parameter for which the resulting method has a considerably small frequency distortion of size \((1/40320)H^ 6\) and also a (slightly) larger interval of periodicity of size 2.75 than the phase-lag of size \((1/12096)H^ 6\) and interval of periodicity of size 2.71 for the implicit method of the authors. More interestingly, it turns out that Noumerov made explicit of the first author [BIT 24, 117- 118 (1984; Zbl 0568.65042)] also has less frequency distortion than the (implicit) Noumerov method.
Reviewer: Z.Jackiewicz

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations
34A34 Nonlinear ordinary differential equations and systems
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brusa, L.; Nigro, L., A one-step method for direct integration of structural dynamic equations, Internat. J. Numer. Methods Engrg., 15, 685-699 (1980) · Zbl 0426.65034
[2] Chawla, M. M.; Rao, P. S., A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems, J. Comput. Appl. Maths., 11, 277-281 (1984) · Zbl 0565.65041
[3] Chawla, M. M., Numerov made explicit has better stability, BIT, 24, 117-118 (1984) · Zbl 0568.65042
[4] Chawla, M. M.; Sharma, S. R., Families of fifth order Nyström methods for \(y\)″ = \(f(x, y)\) and intervals of periodicity, Computing, 26, 247-256 (1981) · Zbl 0437.34035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.